Here, we follow the “hard case”
setup
of [1]}, [2]} for the Lorenz-96 model, with sparse observations in space and time, i.e., \(d = 20\)  (observing every other component of the state) and \(\Delta t_{\rm obs} = 0.4\) . Each observation has independent additive Gaussian noise with variance \(\theta ^2 = 0.5\) .
For reference, we note that a \(\Delta t_{\rm obs}=0.05\)  would
correspond to roughly 6 hours in a global weather
model [3]}, [4]}.
The large time interval between observations makes the
forecast distribution highly non-Gaussian.
As a result, this configuration typically requires ensemble sizes greater than the state dimension \(n\)  to track \(z_{k}^{*}\) . Furthermore, with a high-dimensional state vector it is necessary to tune the map's localization parameters to achieve stable filtering performance over time. As described in Section REF , we tune the localization parameters \(r\)  and \(j\)  for the stochastic map filter and the localization radius for the EnKF, and the inflation parameter \(\zeta \)  for both algorithms. We find that the filters require essentially no forecast inflation for these identical twin experiments.
